SEE the Big Idea. Airport Runway (p. 100) Sculpture (p. 96) City Street t (p. 87) Tiger (p. 75) Guitar (p. 61)

Transcription

1 2 Reasoning and Direct Proofs 2.1 onditional Statements 2.2 Inductive and Deductive Reasoning 2.3 Postulates and Diagrams 2.4 lgebraic Reasoning 2.5 Proving Statements about Segments and ngles 2.6 Proving Geometric Relationships irport Runway (p. 100) Sculpture (p. 96) SEE the ig Idea ity Street t (p. 87) Tiger (p. 75) Guitar (p. 61)

2 Maintaining Mathematical Proficiency Describing a Pattern Example 1 Describe each pattern and write the next two numbers. a. 3, 9, 15, 21,... Each number is 6 more than the previous number. The next two numbers are 27 and 33. b. 3, 9, 27, 81,... Each number is 3 times the previous number. The next two numbers are 243 and 729. Describe the pattern and write the next two numbers , 32, 16, 8, , 20, 50, 125, , 3.4, 4.0, 4.6, , 22, 18, 14, , 1 2, 2 3, 5 6, , 5, 5, 20,... 4 Rewriting Literal Equations Example 2 Solve the literal equation 3x + 6y = 24 for y. 3x + 6y = 24 3x 3x + 6y = 24 3x 6y = 24 3x 6y 6 = 24 3x 6 Write the equation. Subtract 3x from each side. Simplify. Divide each side by 6. y = x Simplify. The rewritten literal equation is y = x. Solve the literal equation for x. 7. 2y 2x = y + 5x = y 5 = 4x y = 8x x 11. y = 4x + zx z = 2x + 6xy 13. STRT RESONING Write the next two terms of the pattern x 4, x 3 y, x 2 y 2, xy 3,.... Explain your reasoning. Dynamic Solutions available at igideasmath.com 57

3 Mathematical Processes Using orrect Reasoning ore oncept Mathematically profi cient students distinguish correct reasoning from fl awed reasoning. Deductive Reasoning When you use deductive reasoning, you start with two or more true statements and deduce or infer the truth of another statement. Here is an example. 1. Premise: If a polygon is a triangle, then the sum of its angle measures is Premise: Polygon is a triangle. 3. onclusion: The sum of the angle measures of polygon is 180. This pattern for deductive reasoning is called a syllogism. Monitoring Progress Recognizing Flawed Reasoning The syllogisms below represent common types of fl awed reasoning. Explain why each conclusion is not valid. a. When it rains, the ground gets wet. b. If is equilateral, then it is isosceles. The ground is wet. is not equilateral. Therefore, it must have rained. Therefore, it must not be isosceles. c. ll squares are polygons. d. No triangles are quadrilaterals. ll trapezoids are quadrilaterals. Some quadrilaterals are not squares. Therefore, all squares are quadrilaterals. Therefore, some squares are not triangles. SOLUTION a. The ground may be wet for another reason. b. triangle can be isosceles but not equilateral. c. ll squares are quadrilaterals, but not because all trapezoids are quadrilaterals. d. No squares are triangles. Decide whether the syllogism represents correct or flawed reasoning. If flawed, explain why the conclusion is not valid. 1. ll triangles are polygons. 2. No trapezoids are rectangles. Figure is a triangle. Some rectangles are not squares. Therefore, figure is a polygon. Therefore, some squares are not trapezoids. 3. If polygon D is a square, then it is a rectangle. 4. If polygon D is a square, then it is a rectangle. Polygon D is a rectangle. Polygon D is not a square. Therefore, polygon D is a square. Therefore, polygon D is not a rectangle. 58 hapter 2 Reasoning and Direct Proofs

4 2.1 onditional Statements Essential Question When is a conditional statement true or false? conditional statement, symbolized by p q, can be written as an if-then statement in which p is the hypothesis and q is the conclusion. Here is an example. If a polygon is a triangle, then the sum of its angle measures is 180. hypothesis, p conclusion, q Determining Whether a Statement Is True or False Work with a partner. hypothesis can either be true or false. The same is true of a conclusion. For a conditional statement to be true, the hypothesis and conclusion do not necessarily both have to be true. Determine whether each conditional statement is true or false. Justify your answer. a. If yesterday was Wednesday, then today is Thursday. b. If an angle is acute, then it has a measure of 30. c. If a month has 30 days, then it is June. d. If an even number is not divisible by 2, then 9 is a perfect cube. ONSTRUTING VILE RGUMENTS To be proficient in math, you need to distinguish correct logic or reasoning from that which is flawed. Work with a partner. Use the points in the coordinate plane to determine whether each statement is true or false. Justify your answer. a. is a right triangle. b. D is an equilateral triangle. c. D is an isosceles triangle. d. Quadrilateral D is a trapezoid. e. Quadrilateral D is a parallelogram. Determining Whether a Statement Is True or False Determining Whether a Statement Is True or False 4 6 x Work with a partner. Determine whether each conditional statement is true or false. Justify your answer. a. If D is a right triangle, then the Pythagorean Theorem is valid for D. b. If and are complementary, then the sum of their measures is 180. c. If figure D is a quadrilateral, then the sum of its angle measures is 180. d. If points,, and are collinear, then they lie on the same line. e. If and D intersect at a point, then they form two pairs of vertical angles y 6 D ommunicate Your nswer 4. When is a conditional statement true or false? 5. Write one true conditional statement and one false conditional statement that are different from those given in Exploration 3. Justify your answer. Section 2.1 onditional Statements 59

5 2.1 Lesson What You Will Learn ore Vocabulary conditional statement, p. 60 if-then form, p. 60 hypothesis, p. 60 conclusion, p. 60 negation, p. 60 converse, p. 61 inverse, p. 61 contrapositive, p. 61 equivalent statements, p. 61 perpendicular lines, p. 62 biconditional statement, p. 63 truth value, p. 64 truth table, p. 64 Write conditional statements. Use definitions written as conditional statements. Write biconditional statements. Make truth tables. Writing onditional Statements ore oncept onditional Statement conditional statement is a logical statement that has two parts, a hypothesis p and a conclusion q. When a conditional statement is written in if-then form, the if part contains the hypothesis and the then part contains the conclusion. Words If p, then q. Symbols p q (read as p implies q ) Rewriting a Statement in If-Then Form Use red to identify the hypothesis and blue to identify the conclusion. Then rewrite the conditional statement in if-then form. a. ll birds have feathers. b. You are in Texas if you are in Houston. SOLUTION a. ll birds have feathers. b. You are in Texas if you are in Houston. If an animal is a bird, then it has feathers. If you are in Houston, then you are in Texas. Monitoring Progress Help in English and Spanish at igideasmath.com Use red to identify the hypothesis and blue to identify the conclusion. Then rewrite the conditional statement in if-then form. 1. ll 30 angles are acute angles. 2. 2x + 7 = 1, because x = 3. ore oncept Negation The negation of a statement is the opposite of the original statement. To write the negation of a statement p, you write the symbol for negation ( ) before the letter. So, not p is written p. Words not p Symbols p Writing a Negation Write the negation of each statement. a. The ball is red. b. The cat is not black. SOLUTION a. The ball is not red. b. The cat is black. 60 hapter 2 Reasoning and Direct Proofs

6 ore oncept Related onditionals onsider the conditional statement below. Words If p, then q. Symbols p q onverse To write the converse of a conditional statement, exchange the hypothesis and the conclusion. Words If q, then p. Symbols q p OMMON ERROR Just because a conditional statement and its contrapositive are both true does not mean that its converse and inverse are both false. The converse and inverse could also both be true. Inverse To write the inverse of a conditional statement, negate both the hypothesis and the conclusion. Words If not p, then not q. Symbols p q ontrapositive To write the contrapositive of a conditional statement, first write the converse. Then negate both the hypothesis and the conclusion. Words If not q, then not p. Symbols q p conditional statement and its contrapositive are either both true or both false. Similarly, the converse and inverse of a conditional statement are either both true or both false. In general, when two statements are both true or both false, they are called equivalent statements. Writing Related onditional Statements Let p be you are a guitar player and let q be you are a musician. Write each statement using symbols. Then decide whether it is true or false. a. If you are a guitar player, then you are a musician. b. If you are a musician, then you are a guitar player. c. If you are not a guitar player, then you are not a musician. d. If you are not a musician, then you are not a guitar player. SOLUTION a. p q true; Guitar players are musicians. b. q p false; Not all musicians play the guitar. c. p q false; Even if you do not play a guitar, you can still be a musician. d. q p true; person who is not a musician cannot be a guitar player. Monitoring Progress In Exercises 3 and 4, write the negation of the statement. Help in English and Spanish at igideasmath.com 3. The shirt is green. 4. The shoes are not red. 5. Let p be the stars are visible and let q be it is night. Write If it is night, then the stars are visible using symbols. Then decide whether it is true or false. Section 2.1 onditional Statements 61

7 Using Definitions You can write a definition as a conditional statement in if-then form or as its converse. oth the conditional statement and its converse are true for definitions. For example, consider the definition of perpendicular lines. If two lines intersect to form a right angle, then they are perpendicular lines. You can also write the definition using the converse: If two lines are perpendicular lines, then they intersect to form a right angle. You can write line is perpendicular to line m as m. m m Using Definitions Decide whether each statement about the diagram is true. Explain your answer using the definitions you have learned. a. D b. E and E are a linear pair. c. E and E are opposite rays. E D SOLUTION a. This statement is true. The right angle symbol in the diagram indicates that the lines intersect to form a right angle. So, you can say the lines are perpendicular. b. This statement is true. y definition, if the noncommon sides of adjacent angles are opposite rays, then the angles are a linear pair. ecause E and E are opposite rays, E and E are a linear pair. c. This statement is false. Point E does not lie on the same line as and, so the rays are not opposite rays. Monitoring Progress Help in English and Spanish at igideasmath.com Use the diagram. Decide whether the statement is true. Explain your answer using the definitions you have learned. F G J M H 6. JMF and FMG are supplementary. 7. Point M is the midpoint of FH. 8. JMF and HMG are vertical angles. 9. FH JG 62 hapter 2 Reasoning and Direct Proofs

8 Writing iconditional Statements ore oncept iconditional Statement When a conditional statement and its converse are both true, you can write them as a single biconditional statement. biconditional statement is a statement that contains the phrase if and only if. Words p if and only if q Symbols p q ny definition can be written as a biconditional statement. Writing a iconditional Statement Rewrite the definition of perpendicular lines as a single biconditional statement. Definition If two lines intersect to form a right angle, then they are perpendicular lines. SOLUTION Let p be two lines intersect to form a right angle and let q be they are perpendicular lines. Use red to identify p and blue to identify q. Write the definition p q. Definition If two lines intersect to form a right angle, then they are perpendicular lines. Write the converse q p. onverse If two lines are perpendicular lines, then they intersect to form a right angle. Use the definition and its converse to write the biconditional statement p q. s s t t iconditional Two lines intersect to form a right angle if and only if they are perpendicular lines. Monitoring Progress Help in English and Spanish at igideasmath.com 10. Rewrite the definition of a right angle as a single biconditional statement. Definition If an angle is a right angle, then its measure is Rewrite the definition of congruent segments as a single biconditional statement. Definition If two line segments have the same length, then they are congruent segments. 12. Rewrite the statements as a single biconditional statement. If Mary is in theater class, then she will be in the fall play. If Mary is in the fall play, then she must be taking theater class. 13. Rewrite the statements as a single biconditional statement. If you can run for President, then you are at least 35 years old. If you are at least 35 years old, then you can run for President. Section 2.1 onditional Statements 63

9 Making Truth Tables The truth value of a statement is either true (T) or false (F). You can determine the conditions under which a conditional statement is true by using a truth table. The truth table below shows the truth values for hypothesis p and conclusion q. onditional p q p q T T T T F F F T T F F T The conditional statement p q is only false when a true hypothesis produces a false conclusion. Two statements are logically equivalent when they have the same truth table. Making a Truth Table Use the truth table above to make truth tables for the converse, inverse, and contrapositive of a conditional statement p q. SOLUTION The truth tables for the converse and the inverse are shown below. Notice that the converse and the inverse are logically equivalent because they have the same truth table. onverse p q q p T T T T F T F T F F F T Inverse p q p q p q T T F F T T F F T T F T T F F F F T T T The truth table for the contrapositive is shown below. Notice that a conditional statement and its contrapositive are logically equivalent because they have the same truth table. ontrapositive p q q p q p T T F F T T F T F F F T F T T F F T T T Monitoring Progress Help in English and Spanish at igideasmath.com 14. Make a truth table for the conditional statement p q. 15. Make a truth table for the conditional statement (p q). 64 hapter 2 Reasoning and Direct Proofs

10 2.1 Exercises Dynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck 1. VOULRY What type of statements are either both true or both false? 2. WHIH ONE DOESN T ELONG? Which statement does not belong with the other three? Explain your reasoning. If today is Tuesday, then tomorrow is Wednesday. If it is Independence Day, then it is July. If an angle is acute, then its measure is less than 90. If you are an athlete, then you play soccer. Monitoring Progress and Modeling with Mathematics In Exercises 3 6, copy the conditional statement. Underline the hypothesis and circle the conclusion. 3. If a polygon is a pentagon, then it has five sides. 4. If two lines form vertical angles, then they intersect. 5. If you run, then you are fast. 6. If you like math, then you like science. In Exercises 7 12, rewrite the conditional statement in if-then form. (See Example 1.) 7. 9x + 5 = 23, because x = Today is Friday, and tomorrow is the weekend. 9. You are in a band, and you play the drums. 10. Two right angles are supplementary angles. 11. Only people who are registered are allowed to vote. 12. The measures of complementary angles sum to 90. b. If the measures of two angles sum to 180, then the two angles are supplementary. c. If the measures of two angles do not sum to 180, then the two angles are not supplementary. d. If two angles are supplementary, then the measures of the angles sum to Let p be the Sun is out and let q be it is daytime. a. If the Sun is out, then it is daytime. b. If it is daytime, then the Sun is out. c. If it is not daytime, then the Sun is not out. d. If the Sun is not out, then it is not daytime. In Exercises 17 20, decide whether the statement about the diagram is true. Explain your answer using the definitions you have learned. (See Example 4.) 17. m = PQ ST P In Exercises 13 and 14, write the negation of the statement. (See Example 2.) S Q T 13. The sky is blue. 14. The lake is cold. In Exercises 15 and 16, write each statement using symbols. Then decide whether each statement is true or false. (See Example 3.) 15. Let p be two angles are supplementary and let q be the measures of the angles sum to m 2 + m 3 = M is the midpoint of. Q M 2 3 N P M a. If two angles are not supplementary, then the measures of the angles do not sum to 180. Section 2.1 onditional Statements 65

11 In Exercises 21 24, rewrite the definition of the term as a biconditional statement. (See Example 5.) 21. The midpoint of a segment is the point that divides the segment into two congruent segments. 22. Two angles are vertical angles when their sides form two pairs of opposite rays. 23. djacent angles are two angles that share a common vertex and side but have no common interior points. 24. Two angles are supplementary angles when the sum of their measures is 180. In Exercises 25 28, rewrite the statements as a single biconditional statement. (See Example 5.) 25. If a polygon has three sides, then it is a triangle. If a polygon is a triangle, then it has three sides. 26. If a polygon has four sides, then it is a quadrilateral. If a polygon is a quadrilateral, then it has four sides. 27. If an angle is a right angle, then it measures 90. If an angle measures 90, then it is a right angle. 28. If an angle is obtuse, then it has a measure between 90 and 180. If an angle has a measure between 90 and 180, then it is obtuse. 29. ERROR NLYSIS Describe and correct the error in rewriting the conditional statement in if-then form. onditional statement ll high school students take four English courses. If-then form If a high school student takes four courses, then all four are English courses. 30. ERROR NLYSIS Describe and correct the error in writing the converse of the conditional statement. onditional statement If it is raining, then I will bring an umbrella. onverse If it is not raining, then I will not bring an umbrella. In Exercises 31 36, create a truth table for the logical statement. (See Example 6.) 31. p q 32. q p 33. ( p q) 34. ( p q) 35. q p 36. (q p) 37. USING STRUTURE The statements below describe three ways that rocks are formed. Igneous rock is formed from the cooling of molten rock. Sedimentary rock is formed from pieces of other rocks. Metamorphic rock is formed by changing temperature, pressure, or chemistry. a. Write each statement in if-then form. b. Write the converse of each of the statements in part (a). Is the converse of each statement true? Explain your reasoning. c. Write a true if-then statement about rocks that is different from the ones in parts (a) and (b). Is the converse of your statement true or false? Explain your reasoning. 38. MKING N RGUMENT Your friend claims the statement If I bought a shirt, then I went to the mall can be written as a true biconditional statement. Your sister says you cannot write it as a biconditional. Who is correct? Explain your reasoning. 39. RESONING You are told that the contrapositive of a statement is true. Will that help you determine whether the statement can be written as a true biconditional statement? Explain your reasoning. 66 hapter 2 Reasoning and Direct Proofs

12 40. PROLEM SOLVING Use the conditional statement to identify the if-then statement as the converse, inverse, or contrapositive of the conditional statement. Then use the symbols to represent both statements. onditional statement If I rode my bike to school, then I did not walk to school. If-then statement If I did not ride my bike to school, then I walked to school. 45. MTHEMTIL ONNETIONS an the statement If x 2 10 = x + 2, then x = 4 be combined with its converse to form a true biconditional statement? 46. RITIL THINKING The largest natural arch in the United States is Landscape rch, located in Thompson, Utah. It spans 290 feet. p q USING STRUTURE In Exercises 41 44, rewrite the conditional statement in if-then form. Then underline the hypothesis and circle the conclusion a. Use the information to write at least two true conditional statements. b. Which type of related conditional statement must also be true? Write the related conditional statements. c. What are the other two types of related conditional statements? Write the related conditional statements. Then determine their truth values. Explain your reasoning. 47. RESONING Which statement has the same meaning as the given statement? Given statement You can watch a movie after you do your homework. 44. D If you do your homework, then you can watch a movie afterward. If you do not do your homework, then you can watch a movie afterward. If you cannot watch a movie afterward, then do your homework. If you can watch a movie afterward, then do not do your homework. 48. THOUGHT PROVOKING Write three conditional statements, where one is always true, one is always false, and one depends on the person interpreting the statement. Section 2.1 onditional Statements 67

13 49. RITIL THINKING One example of a conditional statement involving dates is If today is ugust 31, then tomorrow is September 1. Write a conditional statement using dates from two different months so that the truth value depends on when the statement is read. 50. HOW DO YOU SEE IT? The Venn diagram represents all the musicians at a high school. Write three conditional statements in if-then form describing the relationships between the various groups of musicians. chorus musicians band jazz band 51. MULTIPLE REPRESENTTIONS reate a Venn diagram representing each conditional statement. Write the converse of each conditional statement. Then determine whether each conditional statement and its converse are true or false. Explain your reasoning. a. If you go to the zoo to see a lion, then you will see a cat. b. If you play a sport, then you wear a helmet. c. If this month has 31 days, then it is not February. 52. DRWING ONLUSIONS You measure the heights of your classmates to get a data set. a. Tell whether this statement is true: If x and y are the least and greatest values in your data set, then the mean of the data is between x and y. b. Write the converse of the statement in part (a). Is the converse true? Explain your reasoning. c. opy and complete the statement below using mean, median, or mode to make a conditional statement that is true for any data set. Explain your reasoning. If a data set has a mean, median, and a mode, then the of the data set will always be a data value. 53. WRITING Write a conditional statement that is true, but its converse is false. 54. RITIL THINKING Write a series of if-then statements that allow you to find the measure of each angle, given that m 1 = 90. Use the definition of linear pairs WRITING dvertising slogans such as uy these shoes! They will make you a better athlete! often imply conditional statements. Find an advertisement or write your own slogan. Then write it as a conditional statement. 4 3 Maintaining Mathematical Proficiency Find the pattern. Then draw the next two figures. 56. Reviewing what you learned in previous grades and lessons 57. Find the pattern. Then write the next two numbers , 3, 5, 7, , 23, 34, 45, , 4 3, 8 9, 16 27, , 4, 9, 16, hapter 2 Reasoning and Direct Proofs

14 2.2 Inductive and Deductive Reasoning Essential Question How can you use reasoning to solve problems? conjecture is an unproven statement based on observations. Writing a onjecture Work with a partner. Write a conjecture about the pattern. Then use your conjecture to draw the 10th object in the pattern. a b. ONSTRUTING VILE RGUMENTS To be proficient in math, you need to justify your conclusions and communicate them to others. c. Using a Venn Diagram Work with a partner. Use the Venn diagram to determine whether the statement is true or false. Justify your answer. ssume that no region of the Venn diagram is empty. a. If an item has Property, then it has Property. b. If an item has Property, then it has Property. Property c. If an item has Property, then it has Property. d. Some items that have Property do not have Property. e. If an item has Property, then it does not have Property. f. Some items have both Properties and. g. Some items have both Properties and. Property Property Reasoning and Venn Diagrams Work with a partner. Draw a Venn diagram that shows the relationship between different types of quadrilaterals: squares, rectangles, parallelograms, trapezoids, rhombuses, and kites. Then write several conditional statements that are shown in your diagram, such as If a quadrilateral is a square, then it is a rectangle. ommunicate Your nswer 4. How can you use reasoning to solve problems? 5. Give an example of how you used reasoning to solve a real-life problem. Section 2.2 Inductive and Deductive Reasoning 69

15 2.2 Lesson What You Will Learn ore Vocabulary conjecture, p. 70 inductive reasoning, p. 70 counterexample, p. 71 deductive reasoning, p. 72 Use inductive reasoning. Use deductive reasoning. Using Inductive Reasoning ore oncept Inductive Reasoning conjecture is an unproven statement that is based on observations. You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case. Describing a Visual Pattern Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure. Figure 1 Figure 2 Figure 3 SOLUTION Each circle is divided into twice as many equal regions as the figure number. Sketch the fourth figure by dividing a circle into eighths. Shade the section just above the horizontal segment at the left. Figure 4 Monitoring Progress 1. Sketch the fifth figure in the pattern in Example 1. Sketch the next figure in the pattern. 2. Help in English and Spanish at igideasmath.com hapter 2 Reasoning and Direct Proofs

16 Making and Testing a onjecture Numbers such as 3, 4, and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive integers. SOLUTION Step 1 Find a pattern using a few groups of small numbers = 12 = = 24 = = 33 = = 51 = 17 3 Step 2 Make a conjecture. onjecture The sum of any three consecutive integers is three times the second number. Step 3 Test your conjecture using other numbers. For example, test that it works with the groups 1, 0, 1 and 100, 101, = 0 = = 303 = ore oncept ounterexample To show that a conjecture is true, you must show that it is true for all cases. You can show that a conjecture is false, however, by finding just one counterexample. counterexample is a specific case for which the conjecture is false. Finding a ounterexample student makes the following conjecture about the sum of two numbers. Find a counterexample to disprove the student s conjecture. onjecture The sum of two numbers is always more than the greater number. SOLUTION To find a counterexample, you need to find a sum that is less than the greater number. 2 + ( 3) = ecause a counterexample exists, the conjecture is false. Monitoring Progress Help in English and Spanish at igideasmath.com 4. Make and test a conjecture about the sign of the product of any three negative integers. 5. Make and test a conjecture about the sum of any five consecutive integers. Find a counterexample to show that the conjecture is false. 6. The value of x 2 is always greater than the value of x. 7. The sum of two numbers is always greater than their difference. Section 2.2 Inductive and Deductive Reasoning 71

17 Using Deductive Reasoning ore oncept Deductive Reasoning Deductive reasoning uses facts, definitions, accepted properties, and the laws of logic to form a logical argument. This is different from inductive reasoning, which uses specific examples and patterns to form a conjecture. STUDY TIP valid logical argument may reach a false conclusion if the argument is based on a false hypothesis. TTENDING TO PREISION You can also use the following symbols when writing a logical argument. [ therefore and or Laws of Logic onsider statements p, q, and r. Law of Detachment If p g q is a true statement and statement p is true, then statement q is also true. Law of Syllogism p g q If these statements are true, q g r p g r then this statement is true. Law of ontrapositive If p g q is a true statement and q is a true statement, then p is also a true statement. Determining the Validity of a Logical rgument Write each logical argument symbolically. Then use deductive reasoning to determine whether each argument is valid. a. If two segments are both 5 centimeters long, then the two segments have the same length. Two segments are both 5 centimeters long. Therefore, the two segments have the same length. b. If two segments have the same length, then the two segments are congruent. Two segments are not congruent. Therefore, the two segments do not have the same length. c. If two segments are both 5 centimeters long, then the two segments have the same length. If two segments have the same length, then the two segments are congruent. Therefore, two segments that are both 5 centimeters long are not congruent. SOLUTION Let p be two segments are both 5 centimeters long, let q be two segments have the same length, and let r be two segments are congruent. a. rgument: p g q. Statement p is true. Therefore, statement q is true. p g q is a true statement. y the Law of Detachment, if statement p is true, then statement q is also true. So, the argument is valid. b. rgument: q g r. r is a true statement. Therefore, q is a true statement. q g r is a true statement. y the Law of ontrapositive, if r is a true statement, then q is also a true statement. So, the argument is valid. c. rgument: p g q and q g r. Therefore, p g r is a true statement. p g q is a true statement and q g r is a true statement. y the Law of Syllogism, p g r is a true statement. So, the argument is not valid. 72 hapter 2 Reasoning and Direct Proofs

18 Using Inductive and Deductive Reasoning What conclusion can you make about the product of an even integer and any other integer? SOLUTION MKING SENSE OF PROLEMS In geometry, you will frequently use inductive reasoning to make conjectures. You will also use deductive reasoning to show that conjectures are true or false. You will need to know which type of reasoning to use. Step 1 Look for a pattern in several examples. Use inductive reasoning to make a conjecture. ( 2)(2) = 4 ( 1)(2) = 2 2(2) = 4 3(2) = 6 ( 2)( 4) = 8 ( 1)( 4) = 4 2( 4) = 8 3( 4) = 12 onjecture Even integer ny integer = Even integer Step 2 Let n and m each be any integer. Use deductive reasoning to show that the conjecture is true. 2n is an even integer because any integer multiplied by 2 is even. 2nm represents the product of an even integer 2n and any integer m. 2nm is the product of 2 and an integer nm. So, 2nm is an even integer. The product of an even integer and any integer is an even integer. omparing Inductive and Deductive Reasoning Decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning. a. Each time Monica kicks a ball up in the air, it returns to the ground. So, the next time Monica kicks a ball up in the air, it will return to the ground. b. ll reptiles are cold-blooded. Parrots are not cold-blooded. Sue s pet parrot is not a reptile. SOLUTION a. Inductive reasoning, because a pattern is used to reach the conclusion. b. Deductive reasoning, because facts about animals and the laws of logic are used to reach the conclusion. Monitoring Progress Help in English and Spanish at igideasmath.com 8. Write each logical argument symbolically. Then use deductive reasoning to determine whether each argument is valid. a. If two lines intersect at right angles, then the two lines are perpendicular. Two lines are not perpendicular. Therefore, the two lines intersect at right angles. b. If two lines intersect at right angles, then the two lines are perpendicular. Two lines intersect at right angles. Therefore, the two lines are perpendicular. 9. Use inductive reasoning to make a conjecture about the sum of a number and itself. Then use deductive reasoning to show that the conjecture is true. 10. Decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning. ll multiples of 8 are divisible by is a multiple of 8. So, 64 is divisible by 4. Section 2.2 Inductive and Deductive Reasoning 73

19 2.2 Exercises Dynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck 1. VOULRY How does the prefix counter- help you understand the term counterexample? 2. WRITING Explain the difference between inductive reasoning and deductive reasoning. Monitoring Progress and Modeling with Mathematics In Exercises 3 8, describe the pattern. Then write or draw the next two numbers, letters, or figures. (See Example 1.) 3. 1, 2, 3, 4, 5, , 2, 6, 12, 20, Z, Y, X, W, V, J, F, M,, M, In Exercises 9 12, make and test a conjecture about the given quantity. (See Example 2.) 9. the product of any two even integers 10. the sum of an even integer and an odd integer 11. the quotient of a number and its reciprocal 12. the quotient of two negative integers In Exercises 13 16, find a counterexample to show that the conjecture is false. (See Example 3.) 13. The product of two positive numbers is always greater than either number. 14. If n is a nonzero integer, then n + 1 is always greater n than If two angles are supplements of each other, then one of the angles must be acute. 16. line s divides MN into two line segments. So, the line s is a segment bisector of MN. In Exercises 17 24, write each logical argument symbolically. Then use deductive reasoning to determine whether each argument is valid. (See Example 4.) 17. If a quadrilateral is a square, then it has four right angles. Quadrilateral QRST does not have four right angles. Therefore, quadrilateral QRST is not a square. 18. If a quadrilateral is a square, then it is a rectangle. If a quadrilateral is a rectangle, then it has four right angles. Therefore, a square does not have four right angles. 19. If a point divides a line segment into two congruent line segments, then the point is a midpoint. Point P divides LH into two congruent line segments. Therefore, point P is a midpoint. 20. If a point divides a line segment into two congruent line segments, then the point is a midpoint. Point P is not a midpoint of. Therefore P = P. 21. If a figure is a rhombus, then the figure is a parallelogram. figure is a rhombus. Therefore, the figure is a parallelogram. 22. If x < 2, then x > 2. x < 2. Therefore, x < If a = 3, then 5a = 15. If 5a = 15, then 1 2 a = 1 1 Therefore, if a = 3, then 1 2 a = If x 4 = 81, then x = 3. If x = 3, then x 3 = 27. x 4 = 81. Therefore, x = 3. In Exercises 25 28, state the law of logic that is illustrated. 25. If you do your homework, then you can watch TV. If you watch TV, then you can watch your favorite show. If you do your homework, then you can watch your favorite show hapter 2 Reasoning and Direct Proofs

20 26. If you miss practice the day before a game, then you will not be a starting player in the game. You miss practice on Tuesday. You will not start the game Wednesday. 27. If x > 12, then x + 9 > 20. The value of x is 14. So, x + 9 > If 1 and 2 are vertical angles, then 1 2. If 1 2, then m 1 = m 2. If 1 and 2 are vertical angles, then m 1 = m 2. In Exercises 29 and 30, use inductive reasoning to make a conjecture about the given quantity. Then use deductive reasoning to show that the conjecture is true. (See Example 5.) 29. the sum of two odd integers 30. the product of two odd integers In Exercises 31 34, decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning. (See Example 6.) 31. Each time your mom goes to the store, she buys milk. So, the next time your mom goes to the store, she will buy milk. 32. Rational numbers can be written as fractions. Irrational numbers cannot be written as fractions. So, 1 2 is a rational number. 33. ll men are mortal. Mozart is a man, so Mozart is mortal. V I D E O 37. RESONING The table shows the average weights of several subspecies of tigers. What conjecture can you make about the relation between the weights of female tigers and the weights of male tigers? Explain your reasoning. Weight of female (pounds) Weight of male (pounds) mur engal South hina Sumatran Indo-hinese HOW DO YOU SEE IT? Determine whether you can make each conjecture from the graph. Explain your reasoning. Number of participants (thousands) U.S. High School Girls Lacrosse y x Year 34. Each time you clean your room, you are allowed to go out with your friends. So, the next time you clean your room, you will be allowed to go out with your friends. ERROR NLYSIS In Exercises 35 and 36, describe and correct the error in interpreting the statement. 35. If a figure is a rectangle, then the figure has four sides. trapezoid has four sides. Using the Law of Detachment, you can conclude that a trapezoid is a rectangle. 36. Each day, you get to school before your friend. Using deductive reasoning, you can conclude that you will arrive at school before your friend tomorrow. a. More girls will participate in high school lacrosse in Year 8 than those who participated in Year 7. b. The number of girls participating in high school lacrosse will exceed the number of boys participating in high school lacrosse in Year MTHEMTIL ONNETIONS Use inductive reasoning to write a formula for the sum of the first n positive even integers. 40. FINDING PTTERN The following are the first nine Fibonacci numbers. 1, 1, 2, 3, 5, 8, 13, 21, 34,... a. Make a conjecture about each of the Fibonacci numbers after the first two. b. Write the next three numbers in the pattern. c. Research to find a real-world example of this pattern. Section 2.2 Inductive and Deductive Reasoning 75

21 41. MKING N RGUMENT Which argument is correct? Explain your reasoning. rgument 1: If two angles measure 30 and 60, then the angles are complementary. 1 and 2 are complementary. So, m 1 = 30 and m 2 = 60. rgument 2: If two angles measure 30 and 60, then the angles are complementary. The measure of 1 is 30 and the measure of 2 is 60. So, 1 and 2 are complementary. 42. THOUGHT PROVOKING The first two terms of a sequence are 1 4 and 1. Describe three different possible 2 patterns for the sequence. List the first five terms for each sequence. 43. MTHEMTIL ONNETIONS Use the table to make a conjecture about the relationship between x and y. Then write an equation for y in terms of x. Use the equation to test your conjecture for other values of x. x y RESONING Use the pattern below. Each figure is made of squares that are 1 unit by 1 unit a. Find the perimeter of each figure. Describe the pattern of the perimeters. b. Predict the perimeter of the 20th figure. 45. DRWING ONLUSIONS Decide whether each conclusion is valid. Explain your reasoning. Yellowstone is a national park in Wyoming. You and your friend went camping at Yellowstone National Park. When you go camping, you go canoeing. If you go on a hike, your friend goes with you. You go on a hike. There is a 3-mile-long trail near your campsite. a. You went camping in Wyoming. b. Your friend went canoeing. c. Your friend went on a hike. d. You and your friend went on a hike on a 3-mile-long trail. 46. RITIL THINKING Geologists use the Mohs scale to determine a mineral s hardness. Using the scale, a mineral with a higher rating will leave a scratch on a mineral with a lower rating. Testing a mineral s hardness can help identify the mineral. Mineral Mohs rating Talc Gypsum alcite Fluorite a. The four minerals are randomly labeled,,, and D. Mineral is scratched by Mineral. Mineral is scratched by all three of the other minerals. What can you conclude? Explain your reasoning. b. What additional test(s) can you use to identify all the minerals in part (a)? Maintaining Mathematical Proficiency Determine which postulate is illustrated by the statement. D Reviewing what you learned in previous grades and lessons E = 48. m D = m DE + m E 49. D is the absolute value of the difference of the coordinates of and D. 50. m D is equal to the absolute value of the difference between the real numbers matched with D and on a protractor. 76 hapter 2 Reasoning and Direct Proofs

22 2.3 Postulates and Diagrams Essential Question In a diagram, what can be assumed and what needs to be labeled? Looking at a Diagram Work with a partner. On a piece of paper, draw two perpendicular lines. Label them and D. Look at the diagram from different angles. Do the lines appear perpendicular regardless of the angle at which you look at them? Describe all the angles at which you can look at the lines and have them appear perpendicular. D TTENDING TO PREISION To be proficient in math, you need to state the meanings of the symbols you choose. D view from above Interpreting a Diagram Work with a partner. When you draw a diagram, you are communicating with others. It is important that you include sufficient information in the diagram. Use the diagram to determine which of the following statements you can assume to be true. Explain your reasoning. a. ll the points shown are coplanar. b. Points D, G, and I are collinear. c. Points,, and H are collinear. d. EG and H are perpendicular. e. and D are a linear pair. f. F and D are perpendicular. h. F and D are coplanar. j. F and D intersect. view from upper right g. EG and D are parallel. i. EG and D do not intersect. k. EG and D are perpendicular. l. D and F are vertical angles. m. and FH are the same line. ommunicate Your nswer 3. In a diagram, what can be assumed and what needs to be labeled? 4. Use the diagram in Exploration 2 to write two statements you can assume to be true and two statements you cannot assume to be true. Your statements should be different from those given in Exploration 2. Explain your reasoning. E D F G H I Section 2.3 Postulates and Diagrams 77

23 2.3 Lesson ore Vocabulary line perpendicular to a plane, p. 80 Previous postulate point line plane What You Will Learn Identify postulates using diagrams. Sketch and interpret diagrams. Identifying Postulates Here are seven more postulates involving points, lines, and planes. Postulates Point, Line, and Plane Postulates Postulate 2.1 Two Point Postulate Through any two points, there exists exactly one line. 2.2 Line-Point Postulate line contains at least two points. Example Through points and, there is exactly one line. Line contains at least two points. 2.3 Line Intersection Postulate If two lines intersect, then their intersection is exactly one point. m n The intersection of line m and line n is point. 2.4 Three Point Postulate Through any three noncollinear points, there exists exactly one plane. 2.5 Plane-Point Postulate plane contains at least three noncollinear points. D E F R Through points D, E, and F, there is exactly one plane, plane R. Plane R contains at least three noncollinear points. 2.6 Plane-Line Postulate If two points lie in a plane, then the line containing them lies in the plane. D E F R Points D and E lie in plane R, so DE lies in plane R. 2.7 Plane Intersection Postulate If two planes intersect, then their intersection is a line. S T The intersection of plane S and plane T is line. 78 hapter 2 Reasoning and Direct Proofs

24 Identifying a Postulate Using a Diagram State the postulate illustrated by the diagram. a. If then b. If then SOLUTION a. Line Intersection Postulate If two lines intersect, then their intersection is exactly one point. b. Plane Intersection Postulate If two planes intersect, then their intersection is a line. Identifying Postulates from a Diagram Use the diagram to write examples of the Plane-Point Postulate and the Plane-Line Postulate. Q P m n SOLUTION Plane-Point Postulate Plane P contains at least three noncollinear points,,, and. Plane-Line Postulate Point and point lie in plane P. So, line n containing points and also lies in plane P. Monitoring Progress Help in English and Spanish at igideasmath.com 1. Use the diagram in Example 2. Which postulate allows you to say that the intersection of plane P and plane Q is a line? 2. Use the diagram in Example 2 to write an example of the postulate. a. Two Point Postulate b. Line-Point Postulate c. Line Intersection Postulate Section 2.3 Postulates and Diagrams 79

25 Sketching and Interpreting Diagrams Sketching a Diagram Sketch a diagram showing TV intersecting PQ at point W, so that TW WV. NOTHER WY In Example 3, there are many ways you can sketch the diagram. nother way is shown below. T W P V SOLUTION Step 1 Draw TV and label points T and V. Step 2 Draw point W at the midpoint of TV. Mark the congruent segments. Step 3 Draw PQ through W. t q p P line is a line perpendicular to a plane if and only if the line intersects the plane in a point and is perpendicular to every line in the plane that intersects it at that point. In a diagram, a line perpendicular to a plane must be marked with a right angle symbol, as shown. T W V Q Q Interpreting a Diagram Which of the following statements cannot be assumed from the diagram? T Points,, and F are collinear. Points E,, and D are collinear. plane S D plane T E F D S F intersects at point. SOLUTION No drawn line connects points E,, and D. So, you cannot assume they are collinear. With no right angle marked, you cannot assume D plane T. Monitoring Progress Help in English and Spanish at igideasmath.com Refer back to Example If the given information states that PW and QW are congruent, how can you indicate that in the diagram? 4. Name a pair of supplementary angles in the diagram. Explain. Use the diagram in Example an you assume that plane S intersects plane T at? 6. Explain how you know that. 80 hapter 2 Reasoning and Direct Proofs

26 2.3 Exercises Dynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck 1. OMPLETE THE SENTENE Through any noncollinear points, there exists exactly one plane. 2. WRITING Explain why you need at least three noncollinear points to determine a plane. Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, state the postulate illustrated by the diagram. (See Example 1.) In Exercises 13 20, use the diagram to determine whether you can assume the statement. (See Example 4.) If then If then W Q R N J K L P M X In Exercises 5 8, use the diagram to write an example of the postulate. (See Example 2.) J p K H G 5. Line-Point Postulate (Postulate 2.2) 6. Line Intersection Postulate (Postulate 2.3) 7. Three Point Postulate (Postulate 2.4) q L M 13. Planes W and X intersect at KL. 14. Points K, L, M, and N are coplanar. 15. Points Q, J, and M are collinear. 16. MN and RP intersect. 17. JK lies in plane X. 19. NKL and JKM are vertical angles. 18. PLK is a right angle. 20. NKJ and JKM are supplementary angles. 8. Plane-Line Postulate (Postulate 2.6) In Exercises 9 12, sketch a diagram of the description. (See Example 3.) 9. plane P and line m intersecting plane P at a 90 angle 10. XY in plane P, XY bisected by point, and point not on XY 11. XY intersecting WV at point, so that X = V 12., D, and EF are all in plane P, and point X is the midpoint of all three segments. ERROR NLYSIS In Exercises 21 and 22, describe and correct the error in the statement made about the diagram. 21. M is the midpoint of and D. 22. intersects D at a 90 angle, so D. D M Section 2.3 Postulates and Diagrams 81

27 23. TTENDING TO PREISION Select all the statements about the diagram that you cannot conclude. D S H,, and are coplanar. Plane T intersects plane S in. intersects D. H D. E Plane T plane S. F Point bisects H. G H H and HF are a linear pair. F D. 24. HOW DO YOU SEE IT? Use the diagram of line m and point. Make a conjecture about how many planes can be drawn so that line m and point lie in the same plane. Use postulates to justify your conjecture. m 25. MTHEMTIL ONNETIONS One way to graph a linear equation is to plot two points whose coordinates satisfy the equation and then connect them with a line. Which postulate guarantees this process works for any linear equation? 26. MTHEMTIL ONNETIONS way to solve a system of two linear equations that intersect is to graph the lines and find the coordinates of their intersection. Which postulate guarantees this process works for any two linear equations? F T D In Exercises 27 and 28, (a) rewrite the postulate in if-then form. Then (b) write the converse, inverse, and contrapositive and state which ones are true. 27. Two Point Postulate (Postulate 2.1) 28. Plane-Point Postulate (Postulate 2.5) 29. RESONING hoose the correct symbol to go between the statements. number of points to number of points to determine a line determine a plane < = > 30. RITIL THINKING If two lines intersect, then they intersect in exactly one point by the Line Intersection Postulate (Postulate 2.3). Do the two lines have to be in the same plane? Draw a picture to support your answer. Then explain your reasoning. 31. MKING N RGUMENT Your friend claims that even though two planes intersect in a line, it is possible for three planes to intersect in a point. Is your friend correct? Explain your reasoning. 32. MKING N RGUMENT Your friend claims that by the Plane Intersection Postulate (Post. 2.7), any two planes intersect in a line. Is your friend s interpretation of the Plane Intersection Postulate (Post. 2.7) correct? Explain your reasoning. 33. STRT RESONING Points E, F, and G all lie in plane P and in plane Q. What must be true about points E, F, and G so that planes P and Q are different planes? What must be true about points E, F, and G to force planes P and Q to be the same plane? Make sketches to support your answers. 34. THOUGHT PROVOKING The postulates in this book represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. line is a circle on the sphere whose diameter is equal to the diameter of the sphere. plane is the surface of the sphere. Find a postulate on page 78 that is not true in spherical geometry. Explain your reasoning. Maintaining Mathematical Proficiency Solve the equation. Tell which algebraic property of equality you used. 35. t 6 = x = x = Reviewing what you learned in previous grades and lessons x 7 = 5 82 hapter 2 Reasoning and Direct Proofs

28 2.4 lgebraic Reasoning Essential Question Essential Question How can algebraic properties help you solve an equation? Justifying Steps in a Solution Work with a partner. In previous courses, you studied different properties, such as the properties of equality and the Distributive, ommutative, and ssociative Properties. Write the property that justifies each of the following solution steps. lgebraic Step Justification 2(x + 3) 5 = 5x + 4 Write given equation. 2x = 5x + 4 2x + 1 = 5x + 4 2x 2x + 1 = 5x 2x = 3x = 3x = 3x 3 3 = 3x 3 1 = x x = 1 LOOKING FOR STRUTURE To be proficient in math, you need to look closely to discern a pattern or structure. Stating lgebraic Properties Work with a partner. The symbols and represent addition and multiplication (not necessarily in that order). Determine which symbol represents which operation. Justify your answer. Then state each algebraic property being illustrated. Example of Property Name of Property 5 6 = = (5 6) = (4 5) 6 4 (5 6) = (4 5) = = = 5 4 (5 6) = ommunicate Your nswer 3. How can algebraic properties help you solve an equation? 4. Solve 3(x + 1) 1 = 13. Justify each step. Section 2.4 lgebraic Reasoning 83

29 2.4 Lesson What You Will Learn ore Vocabulary Previous equation solve an equation formula Use lgebraic Properties of Equality to justify the steps in solving an equation. Use the Distributive Property to justify the steps in solving an equation. Use properties of equality involving segment lengths and angle measures. Using lgebraic Properties of Equality When you solve an equation, you use properties of real numbers. Segment lengths and angle measures are real numbers, so you can also use these properties to write logical arguments about geometric figures. ore oncept lgebraic Properties of Equality Let a, b, and c be real numbers. ddition Property of Equality If a = b, then a + c = b + c. Subtraction Property of Equality If a = b, then a c = b c. Multiplication Property of Equality If a = b, then a c = b c, c 0. Division Property of Equality If a = b, then a c = b c, c 0. Substitution Property of Equality If a = b, then a can be substituted for b (or b for a) in any equation or expression. REMEMER Inverse operations undo each other. ddition and subtraction are inverse operations. Multiplication and division are inverse operations. Justifying Steps Solve 3x + 2 = 23 4x. Justify each step. SOLUTION Equation Explanation Reason 3x + 2 = 23 4x Write the equation. Given 3x x = 23 4x + 4x dd 4x to each side. ddition Property of Equality 7x + 2 = 23 ombine like terms. Simplify. 7x = 23 2 Subtract 2 from each side. Subtraction Property of Equality 7x = 21 ombine constant terms. Simplify. x = 3 Divide each side by 7. Division Property of Equality The solution is x = 3. Monitoring Progress Help in English and Spanish at igideasmath.com Solve the equation. Justify each step. 1. 6x 11 = p 9 = 10p z = 1 + 5z 84 hapter 2 Reasoning and Direct Proofs

30 Using the Distributive Property ore oncept Distributive Property Let a, b, and c be real numbers. Sum a(b + c) = ab + ac Difference a(b c) = ab ac Using the Distributive Property Solve 5(7w + 8) = 30. Justify each step. SOLUTION Equation Explanation Reason 5(7w + 8) = 30 Write the equation. Given 35w 40 = 30 Multiply. Distributive Property 35w = 70 dd 40 to each side. ddition Property of Equality w = 2 Divide each side by 35. Division Property of Equality The solution is w = 2. Solving a Real-Life Problem You get a raise at your part-time job. To write your raise as a percent, use the formula p(r + 1) = n, where p is your previous wage, r is the percent increase (as a decimal), and n is your new wage. Solve the formula for r. What is your raise written as a percent when your hourly wage increases from $7.25 to $7.54 per hour? REMEMER When evaluating expressions, use the order of operations. SOLUTION Step 1 Solve for r in the formula p(r + 1) = n. Equation Explanation Reason p(r + 1) = n Write the equation. Given pr + p = n Multiply. Distributive Property pr = n p Subtract p from each side. Subtraction Property of Equality r = n p Divide each side by p. Division Property of Equality p Step 2 Evaluate r = n p when n = 7.54 and p = p r = n p = = 0.29 p = 0.04 Your raise is 4%. Monitoring Progress Solve the equation. Justify each step. Help in English and Spanish at igideasmath.com 4. 3(3x + 14) = = 10b + 6(2 b) 6. Solve the formula = 1 bh for b. Justify each step. Then find the base of a 2 triangle whose area is 952 square feet and whose height is 56 feet. Section 2.4 lgebraic Reasoning 85

31 Using Other Properties of Equality The following properties of equality are true for all real numbers. Segment lengths and angle measures are real numbers, so these properties of equality are true for all segment lengths and angle measures. ore oncept Reflexive, Symmetric, and Transitive Properties of Equality Real Numbers Segment Lengths ngle Measures Reflexive Property a = a = m = m Symmetric Property If a = b, then b = a. If = D, then D =. If m = m, then m = m. Transitive Property If a = b and b = c, then a = c. If = D and D = EF, then = EF. If m = m and m = m, then m = m. You reflect the beam of a spotlight off a mirror lying flat on a stage, as shown. Determine whether m D = m E. Using Properties of Equality with ngle Measures D E SOLUTION Equation Explanation Reason m 1 = m 3 Marked in diagram. Given m D = m 3 + m 2 dd measures of ngle ddition Postulate (Post. 1.4) adjacent angles. m D = m 1 + m 2 Substitute m 1 Substitution Property of Equality for m 3. m 1 + m 2 = m E dd measures of ngle ddition Postulate (Post. 1.4) adjacent angles. m D = m E oth measures are Transitive Property of Equality equal to the sum m 1 + m 2. Monitoring Progress Help in English and Spanish at igideasmath.com Name the property of equality that the statement illustrates. 7. If m 6 = m 7, then m 7 = m = m 1 = m 2 and m 2 = m 5. So, m 1 = m hapter 2 Reasoning and Direct Proofs

32 Modeling with Mathematics park, a shoe store, a pizza shop, and a movie theater are located in order on a city street. The distance between the park and the shoe store is the same as the distance between the pizza shop and the movie theater. Show that the distance between the park and the pizza shop is the same as the distance between the shoe store and the movie theater. SOLUTION 1. Understand the Problem You know that the locations lie in order and that the distance between two of the locations (park and shoe store) is the same as the distance between the other two locations (pizza shop and movie theater). You need to show that two of the other distances are the same. 2. Make a Plan Draw and label a diagram to represent the situation. park shoe store pizza shop movie theater Modify your diagram by letting the points P, S, Z, and M represent the park, the shoe store, the pizza shop, and the movie theater, respectively. Show any mathematical relationships. P S Z M Use the Segment ddition Postulate (Postulate 1.2) to show that PZ = SM. 3. Solve the Problem Equation Explanation Reason PS = ZM Marked in diagram. Given PZ = PS + SZ dd lengths of Segment ddition Postulate (Post. 1.2) adjacent segments. SM = SZ + ZM dd lengths of Segment ddition Postulate (Post. 1.2) adjacent segments. PS + SZ = ZM + SZ dd SZ to each side ddition Property of Equality of PS = ZM. PZ = SM Substitute PZ for Substitution Property of Equality PS + SZ and SM for ZM + SZ. 4. Look ack Reread the problem. Make sure your diagram is drawn precisely using the given information. heck the steps in your solution. Monitoring Progress Help in English and Spanish at igideasmath.com Name the property of equality that the statement illustrates. 10. If JK = KL and KL = 16, then JK = PQ = ST, so ST = PQ. 12. ZY = ZY 13. In Example 5, a hot dog stand is located halfway between the shoe store and the pizza shop, at point H. Show that PH = HM. Section 2.4 lgebraic Reasoning 87

33 2.4 Exercises Dynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck 1. VOULRY The statement The measure of an angle is equal to itself is true because of what property? 2. DIFFERENT WORDS, SME QUESTION Which is different? Find both answers. What property justifies the following statement? If c = d, then d = c. If JK = LM, then LM = JK. If e = f and f = g, then e = g. If m R = m S, then m S = m R. Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, write the property that justifies each step. 3. 3x 12 = 7x + 8 Given 4x 12 = 8 4x = 20 x = (x 1) = 4x + 13 Given 5x 5 = 4x + 13 x 5 = 13 x = 18 In Exercises 5 14, solve the equation. Justify each step. (See Examples 1 and 2.) 5. 5x 10 = x + 17 = x 8 = 6x x + 9 = 16 3x 9. 5(3x 20) = (2x + 11) = ( x 5) = (3x + 4) = 18x 13. 4(5x 9) = 2(x + 7) 14. 3(4x + 7) = 5(3x + 3) In Exercises 15 20, solve the equation for y. Justify each step. (See Example 3.) 15. 5x + y = x + 2y = y + 0.5x = x 3 4 y = y = 30x x + 7 = 7 + 9y In Exercises 21 24, solve the equation for the given variable. Justify each step. (See Example 3.) 21. = 2πr ; r 22. I = Prt; P 23. S = 180(n 2); n 24. S = 2πr 2 + 2πrh; h In Exercises 25 32, name the property of equality that the statement illustrates. 25. If x = y, then 3x = 3y. 26. If M = M, then M + 5 = M x = x 28. If x = y, then y = x. 29. m Z = m Z 30. If m = 29 and m = 29, then m = m. 31. If = LM, then LM =. 32. If = XY and XY = 8, then = hapter 2 Reasoning and Direct Proofs

34 In Exercises 33 40, use the property to copy and complete the statement. 33. Substitution Property of Equality: If = 20, then + D =. 34. Symmetric Property of Equality: If m 1 = m 2, then. 35. ddition Property of Equality: If = D, then + EF =. 36. Multiplication Property of Equality: If = D, then 5 =. 37. Subtraction Property of Equality: If LM = XY, then LM GH =. 38. Distributive Property: If 5(x + 8) = 2, then + = Transitive Property of Equality: If m 1 = m 2 and m 2 = m 3, then. 40. Reflexive Property of Equality: m =. ERROR NLYSIS In Exercises 41 and 42, describe and correct the error in solving the equation x = x x = 24 x = 3 6x + 14 = 32 6x = 18 x = 3 Given ddition Property of Equality Division Property of Equality Given Division Property of Equality Simplify. 43. REWRITING FORMUL The formula for the perimeter P of a rectangle is P = 2 + 2w, where is the length and w is the width. Solve the formula for. Justify each step. Then find the length of a rectangular lawn with a perimeter of 32 meters and a width of 5 meters. 44. REWRITING FORMUL The formula for the area of a trapezoid is = 1 2 h (b 1 + b 2 ), where h is the height and b 1 and b 2 are the lengths of the two bases. Solve the formula for b 1. Justify each step. Then find the length of one of the bases of the trapezoid when the area of the trapezoid is 91 square meters, the height is 7 meters, and the length of the other base is 20 meters. 45. NLYZING RELTIONSHIPS In the diagram, m D = m E. Show that m 1 = m 3. (See Example 4.) NLYZING RELTIONSHIPS In the diagram, = D. Show that = D. (See Example 5.) E D 47. NLYZING RELTIONSHIPS opy and complete the table to show that m 2 = m 3. E Equation m 1 = m 4, m EHF = 90, m GHF = 90 m EHF = m GHF m EHF = m 1 + m 2 m GHF = m 3 + m 4 m 1 + m 2 = m 3 + m 4 m 2 = m H F 4 Given G Reason D Substitution Property of Equality 48. WRITING ompare the Reflexive Property of Equality with the Symmetric Property of Equality. How are the properties similar? How are they different? Section 2.4 lgebraic Reasoning 89

35 RESONING In Exercises 49 and 50, show that the perimeter of is equal to the perimeter of D D D 51. MTHEMTIL ONNETIONS In the figure, ZY XW, ZX = 5x + 17, YW = 10 2x, and YX = 3. Find ZY and XW. Z V Y X 52. HOW DO YOU SEE IT? The bar graph shows the number of hours each employee works at a grocery store. Give an example of the Reflexive, Symmetric, and Transitive Properties of Equality. Hours Hours Worked Employee 6 7 W 53. TTENDING TO PREISION Which of the following statements illustrate the Symmetric Property of Equality? Select all that apply. If = RS, then RS =. If x = 9, then 9 = x. If D =, then D =. D = E If = LM and LM = RT, then = RT. F If XY = EF, then FE = XY. 54. THOUGHT PROVOKING Write examples from your everyday life to help you remember the Reflexive, Symmetric, and Transitive Properties of Equality. Justify your answers. 55. MULTIPLE REPRESENTTIONS The formula to convert a temperature in degrees Fahrenheit ( F) to degrees elsius ( ) is = 5 (F 32). 9 a. Solve the formula for F. Justify each step. b. Make a table that shows the conversion to Fahrenheit for each temperature: 0, 20, 32, and 41. c. Use your table to graph the temperature in degrees Fahrenheit as a function of the temperature in degrees elsius. Is this a linear function? 56. RESONING Select all the properties that would also apply to inequalities. Explain your reasoning. ddition Property Subtraction Property Substitution Property D Reflexive Property E Symmetric Property F Transitive Property Maintaining Mathematical Proficiency Name the definition, property, or postulate that is represented by each diagram. Reviewing what you learned in previous grades and lessons 57. X 58. M Y D Z L P N m D + m D = m XY + YZ = XZ 90 hapter 2 Reasoning and Direct Proofs

36 2.5 Proving Statements about Segments and ngles Essential Question How can you prove a mathematical statement? proof is a logical argument that uses deductive reasoning to show that a statement is true. Writing Reasons in a Proof RESONING STRTLY To be proficient in math, you need to know and be able to use algebraic properties. Work with a partner. Four steps of a proof are shown. Write the reasons for each statement. Given = + Prove = STTEMENTS RESONS 1. = + 1. Given 2. + = = = 4. Writing Steps in a Proof Work with a partner. Six steps of a proof are shown. omplete the statements that correspond to each reason. Given m 1 = m 3 E D Prove m E = m D STTEMENTS RESONS Given 2. m E = m 2 + m 3 2. ngle ddition Postulate (Post.1.4) 3. m E = m 2 + m 1 3. Substitution Property of Equality 4. m E = 4. ommutative Property of ddition 5. m 1 + m 2 = 5. ngle ddition Postulate (Post.1.4) Transitive Property of Equality ommunicate Your nswer 3. How can you prove a mathematical statement? 4. Use the given information and the figure to write a proof for the statement. Given is the midpoint of. is the midpoint of D. D Prove = D Section 2.5 Proving Statements about Segments and ngles 91

37 2.5 Lesson What You Will Learn ore Vocabulary proof, p. 92 two-column proof, p. 92 theorem, p. 93 Write two-column proofs. Name and prove properties of congruence. Writing Two-olumn Proofs proof is a logical argument that uses deductive reasoning to show that a statement is true. There are several formats for proofs. two-column proof has numbered statements and corresponding reasons that show an argument in a logical order. In a two-column proof, each statement in the left-hand column is either given information or the result of applying a known property or fact to statements already made. Each reason in the right-hand column is the explanation for the corresponding statement. Writing a Two-olumn Proof Write a two-column proof for the situation in Example 4 from the Section 2.4 lesson. Given m l = m 3 Prove m D = m E D E STTEMENTS RESONS 1. m 1 = m 3 1. Given 2. m D = m 3 + m 2 2. ngle ddition Postulate (Post.1.4) 3. m D = m 1 + m 2 3. Substitution Property of Equality 4. m 1 + m 2 = m E 4. ngle ddition Postulate (Post.1.4) 5. m D = m E 5. Transitive Property of Equality Monitoring Progress Help in English and Spanish at igideasmath.com 1. Six steps of a two-column proof are shown. opy and complete the proof. Given T is the midpoint of SU. S 7x T 3x + 20 U Prove x = 5 STTEMENTS 1. T is the midpoint of SU. 2. ST TU RESONS Definition of midpoint 3. ST = TU 3. Definition of congruent segments 4. 7x = 3x Subtraction Property of Equality 6. x = hapter 2 Reasoning and Direct Proofs

38 Using Properties of ongruence The reasons used in a proof can include definitions, properties, postulates, and theorems. theorem is a statement that can be proven. Once you have proven a theorem, you can use the theorem as a reason in other proofs. Theorems Theorem 2.1 Properties of Segment ongruence Segment congruence is reflexive, symmetric, and transitive. Reflexive For any segment,. Symmetric If D, then D. Transitive If D and D EF, then EF. Proofs Ex. 11, p. 95; Example 3, p. 93; hapter Review 2.5 Example, p. 109 Theorem 2.2 Properties of ngle ongruence ngle congruence is reflexive, symmetric, and transitive. Reflexive For any angle,. Symmetric If, then. Transitive If and, then. Proofs Ex. 25, p. 109; 2.5 oncept Summary, p. 94; Ex. 12, p. 95 Naming Properties of ongruence Name the property that the statement illustrates. a. If T V and V R, then T R. b. If JL YZ, then YZ JL. SOLUTION a. Transitive Property of ngle ongruence b. Symmetric Property of Segment ongruence STUDY TIP When writing a proof, organize your reasoning by copying or drawing a diagram for the situation described. Then identify the Given and Prove statements. In this lesson, most of the proofs involve showing that congruence and equality are equivalent. You may find that what you are asked to prove seems to be obviously true. It is important to practice writing these proofs to help you prepare for writing more-complicated proofs in later chapters. Proving a Symmetric Property of ongruence Write a two-column proof for the Symmetric Property of Segment ongruence. Given LM NP L M N P Prove NP LM STTEMENTS 1. LM NP RESONS 1. Given 2. LM = NP 2. Definition of congruent segments 3. NP = LM 3. Symmetric Property of Equality 4. NP LM 4. Definition of congruent segments Section 2.5 Proving Statements about Segments and ngles 93

39 Writing a Two-olumn Proof Prove this property of midpoints: If you know that M is the midpoint of, prove that is two times M and M is one-half. Given M is the midpoint of. Prove = 2M, M = 1 2 STTEMENTS 1. M is the midpoint of. 2. M M RESONS 1. Given M 2. Definition of midpoint 3. M = M 3. Definition of congruent segments 4. M + M = 4. Segment ddition Postulate (Post. 1.2) 5. M + M = 5. Substitution Property of Equality 6. 2M = 6. Distributive Property 7. M = 1 7. Division Property of Equality 2 Monitoring Progress Name the property that the statement illustrates. 2. GH GH Help in English and Spanish at igideasmath.com 3. If K P, then P K. 4. Look back at Example 4. What would be different if you were proving that = 2 M and that M = 1 instead? 2 oncept Summary Writing a Two-olumn Proof In a proof, you make one statement at a time until you reach the conclusion. ecause you make statements based on facts, you are using deductive reasoning. Usually the first statement-and-reason pair you write is given information. Proof of the Symmetric Property of ngle ongruence statements based on facts that you know or on conclusions from deductive reasoning Given 1 2 Prove 2 1 STTEMENTS RESONS Given 2. m 1 = m 2 2. Definition of congruent angles 3. m 2 = m 1 3. Symmetric Property of Equality Definition of congruent angles The number of statements will vary. Remember to give a reason for the last statement. 1 opy or draw diagrams and label given information to help develop proofs. Do not mark or label the information in the Prove statement on the diagram. 2 definitions, postulates, or proven theorems that allow you to state the corresponding statement 94 hapter 2 Reasoning and Direct Proofs

40 2.5 Exercises Dynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck 1. WRITING How is a theorem different from a postulate? 2. OMPLETE THE SENTENE In a two-column proof, each is on the left and each is on the right. Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, copy and complete the proof. (See Example 1.) 3. Given PQ = RS Prove PR = QS P Q R S STTEMENTS RESONS 1. PQ = RS PQ + QR = RS + QR Segment ddition Postulate (Post. 1.2) 4. RS + QR = QS 4. Segment ddition Postulate (Post. 1.2) 5. PR = QS Given 1 is a complement of Prove 1 is a complement of STTEMENTS RESONS 1. 1 is a complement of Given m 1 + m 2 = m 2 = m 3 4. Definition of congruent angles Substitution Property of Equality 6. 1 is a complement of In Exercises 5 10, name the property that the statement illustrates. (See Example 2.) 5. If PQ ST and ST UV, then PQ UV. 6. F F 7. If G H, then H G. 8. DE DE 9. If XY UV, then UV XY. 10. If L M and M N, then L N. PROOF In Exercises 11 and 12, write a two-column proof for the property. (See Example 3.) 11. Reflexive Property of Segment ongruence (Thm. 2.1) 12. Transitive Property of ngle ongruence (Thm. 2.2) PROOF In Exercises 13 and 14, write a two-column proof. (See Example 4.) 13. Given GFH GHF Prove EFG and GHF are supplementary. E F 14. Given FG, F bisects and DG. Prove DF G H D G F Section 2.5 Proving Statements about Segments and ngles 95

41 15. ERROR NLYSIS In the diagram, MN LQ and LQ PN. Describe and correct the error in the reasoning. ecause MN LQ and LQ PN L, then MN PN by the Reflexive Property of Segment Q ongruence (Thm. 2.1). P M N 19. WRITING Explain why you do not use inductive reasoning when writing a proof. 20. HOW DO YOU SEE IT? Use the figure to write Given and Prove statements for each conclusion. J N M K L 16. MODELING WITH MTHEMTIS The distance from the restaurant to the shoe store is the same as the distance from the café to the florist. The distance from the shoe store to the movie theater is the same as the distance from the movie theater to the cafe, and from the florist to the dry cleaners. SHOE STORE restaurant shoe store movie theater Flowers DRY LENERS café florist dry cleaners Use the steps below to prove that the distance from the restaurant to the movie theater is the same as the distance from the café to the dry cleaners. a. State what is given and what is to be proven for the situation. b. Write a two-column proof. 17. RESONING In the sculpture shown, 1 2 and 2 3. lassify the triangle and justify your answer. 18. MKING N RGUMENT In the figure, SR and QR. Your friend claims that, because of this, by the Transitive Property of Segment Q ongruence (Thm. 2.1). Is your friend correct? Explain your reasoning. S R a. The acute angles of a right triangle are complementary. b. segment connecting the midpoints of two sides of a triangle is half as long as the third side. 21. RESONING Fold two corners of a piece of paper so their edges match, as shown. a. What do you notice about the angle formed at the top of the page by the folds? b. Write a two-column proof to show that the angle measure is always the same no matter how you make the folds. 22. THOUGHT PROVOKING The distance from Springfield to Lakewood ity is equal to the distance from Springfield to ettsville. Janisburg is 50 miles farther from Springfield than ettsville. Moon Valley is 50 miles farther from Springfield than Lakewood ity is. Use line segments to draw a diagram that represents this situation. 23. MTHEMTIL ONNETIONS Solve for x using the given information. Justify each step. Given QR PQ, RS PQ P Q 2x + 5 R 1 1 S 10 ] 3x 2 2 Maintaining Mathematical Proficiency Use the figure is a complement of 4, is a supplement of 2, and m 1 = 33. Find m 4. and m 2 = 147. Find m Name a pair of vertical angles. Reviewing what you learned in previous grades and lessons hapter 2 Reasoning and Direct Proofs

42 2.6 Proving Geometric Relationships Essential Question How can you use a flowchart to prove a mathematical statement? Matching Reasons in a Flowchart Proof Work with a partner. Match each reason with the correct step in the flowchart. MODELING WITH MTHEMTIS To be proficient in math, you need to map relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. Given = + Prove = = + + = + = + =. Segment ddition Postulate (Post. 1.2). Given. Transitive Property of Equality D. Subtraction Property of Equality Matching Reasons in a Flowchart Proof Work with a partner. Match each reason with the correct step in the flowchart. Given m 1 = m 3 Prove m E = m D m 1 = m 3 E D m E = m 2 + m 3 m E = m 2 + m 1 m E = m 1 + m 2 m 1 + m 2 = m D m E = m D. ngle ddition Postulate (Post. 1.4). Transitive Property of Equality. Substitution Property of Equality D. ngle ddition Postulate (Post. 1.4) E. Given F. ommutative Property of ddition ommunicate Your nswer 3. How can you use a flowchart to prove a mathematical statement? 4. ompare the flowchart proofs above with the two-column proofs in the Section 2.5 Explorations. Explain the advantages and disadvantages of each. Section 2.6 Proving Geometric Relationships 97

43 2.6 Lesson What You Will Learn ore Vocabulary flowchart proof, or flow proof, p. 98 paragraph proof, p. 100 STUDY TIP When you prove a theorem, write the hypothesis of the theorem as the Given statement. The conclusion is what you must Prove. Write flowchart proofs to prove geometric relationships. Write paragraph proofs to prove geometric relationships. Writing Flowchart Proofs nother proof format is a flowchart proof, or flow proof, which uses boxes and arrows to show the flow of a logical argument. Each reason is below the statement it justifies. flowchart proof of the Right ngles ongruence Theorem is shown in Example 1. This theorem is useful when writing proofs involving right angles. Theorem Theorem 2.3 Right ngles ongruence Theorem ll right angles are congruent. Proof Example 1, p. 98 Proving the Right ngles ongruence Theorem Use the given flowchart proof to write a two-column proof of the Right ngles ongruence Theorem. Given 1 and 2 are right angles. Prove 1 2 Flowchart Proof and 2 are right angles. Given m 1 = 90, m 2 = 90 m l = m 2 l 2 Definition of right angle Two-olumn Proof STTEMENTS Transitive Property of Equality RESONS 1. 1 and 2 are right angles. 1. Given Definition of congruent angles 2. m 1 = 90, m 2 = Definition of right angle 3. m 1 = m 2 3. Transitive Property of Equality Definition of congruent angles Monitoring Progress 1. opy and complete the flowchart proof. Then write a two-column proof. Given, D Prove, D Help in English and Spanish at igideasmath.com D Given Definition of lines 98 hapter 2 Reasoning and Direct Proofs

44 Theorems Theorem 2.4 ongruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles), then they are congruent. If 1 and 2 are supplementary and 3 and 2 are supplementary, then Proof Example 2, p. 99 (case 1); Ex. 20, p. 105 (case 2) Theorem 2.5 ongruent omplements Theorem If two angles are complementary to the same angle (or to congruent angles), then they are congruent. If 4 and 5 are complementary and 6 and 5 are complementary, then 4 6. Proof Ex. 19, p. 104 (case 1); Ex. 22, p. 105 (case 2) To prove the ongruent Supplements Theorem, you must prove two cases: one with angles supplementary to the same angle and one with angles supplementary to congruent angles. The proof of the ongruent omplements Theorem also requires two cases. Proving a ase of ongruent Supplements Theorem Use the given two-column proof to write a flowchart proof that proves that two angles supplementary to the same angle are congruent. Given 1 and 2 are supplementary. 3 and 2 are supplementary. 3 Prove Two-olumn Proof STTEMENTS 1. 1 and 2 are supplementary. 3 and 2 are supplementary. RESONS 1. Given 2. m 1 + m 2 = 180, m 3 + m 2 = Definition of supplementary angles 3. m 1 + m 2 = m 3 + m 2 3. Transitive Property of Equality 4. m 1 = m 3 4. Subtraction Property of Equality Definition of congruent angles Flowchart Proof 1 and 2 are supplementary. Given 3 and 2 are supplementary. Given m 1 + m 2 = 180 Definition of supplementary angles m 3 + m 2 = 180 Definition ii of supplementary angles m 1 + m 2 = m 3 + m 2 Transitive Property of Equality m 1 = m 3 Subtraction Property of Equality 1 3 Definition i of congruent angles Section 2.6 Proving Geometric Relationships 99

45 Writing Paragraph Proofs nother proof format is a paragraph proof, which presents the statements and reasons of a proof as sentences in a paragraph. It uses words to explain the logical flow of the argument. Two intersecting lines form pairs of vertical angles and linear pairs. The Linear Pair Postulate formally states the relationship between linear pairs. You can use this postulate to prove the Vertical ngles ongruence Theorem. Postulate and Theorem Postulate 2.8 Linear Pair Postulate If two angles form a linear pair, then they are supplementary. 1 and 2 form a linear pair, so 1 and 2 are supplementary and m l + m 2 = Theorem 2.6 Vertical ngles ongruence Theorem Vertical angles are congruent Proof Example 3, p , 2 4 Use the given paragraph proof to write a two-column proof of the Vertical ngles ongruence Theorem. Proving the Vertical ngles ongruence Theorem STUDY TIP In paragraph proofs, transitional words such as so, then, and therefore help make the logic clear. JUSTIFYING STEPS You can use information labeled in a diagram in your proof. Given 5 and 7 are vertical angles. Prove 5 7 Paragraph Proof 5 and 7 are vertical angles formed by intersecting lines. s shown in the diagram, 5 and 6 are a linear pair, and 6 and 7 are a linear pair. Then, by the Linear Pair Postulate, 5 and 6 are supplementary and 6 and 7 are supplementary. So, by the ongruent Supplements Theorem, 5 7. Two-olumn Proof STTEMENTS RESONS 1. 5 and 7 are vertical angles. 1. Given 2. 5 and 6 are a linear pair. 6 and 7 are a linear pair and 6 are supplementary. 6 and 7 are supplementary. 2. Definition of linear pair, as shown in the diagram 3. Linear Pair Postulate ongruent Supplements Theorem 100 hapter 2 Reasoning and Direct Proofs

46 Monitoring Progress Help in English and Spanish at igideasmath.com 2. opy and complete the two-column proof. Then write a flowchart proof. Given = DE, = D Prove E D E STTEMENTS RESONS 1. = DE, = D 1. Given 2. + = + DE 2. ddition Property of Equality Substitution Property of Equality 4. + =, D + DE = E Substitution Property of Equality 6. E Rewrite the two-column proof in Example 3 without using the ongruent Supplements Theorem. How many steps do you save by using the theorem? Using ngle Relationships Find the value of x. Q SOLUTION T (3x + 1) TPS and QPR are vertical angles. y the Vertical ngles ongruence Theorem, the angles 148 P are congruent. Use this fact to write and solve S an equation. m TPS = m QPR Definition of congruent angles 148 = (3x + 1) Substitute angle measures. 147 = 3x Subtract 1 from each side. 49 = x Divide each side by 3. R So, the value of x is 49. Monitoring Progress Use the diagram and the given angle measure to find the other three angle measures. 4. m 1 = m 2 = m 4 = 88 Help in English and Spanish at igideasmath.com Find the value of w. (5w + 3) 98 Section 2.6 Proving Geometric Relationships 101

47 Using the Vertical ngles ongruence Theorem Write a paragraph proof. Given 1 4 Prove Paragraph Proof 1 and 4 are congruent. y the Vertical ngles ongruence Theorem, 1 2 and 3 4. y the Transitive Property of ngle ongruence (Theorem 2.2), 2 4. Using the Transitive Property of ngle ongruence (Theorem 2.2) once more, 2 3. Monitoring Progress Help in English and Spanish at igideasmath.com 8. Write a paragraph proof. Given 1 is a right angle. Prove 2 is a right angle. 2 1 oncept Summary Types of Proofs Symmetric Property of ngle ongruence (Theorem 2.2) Given 1 2 Prove 2 1 Two-olumn Proof STTEMENTS RESONS Given m 1 = m 2 2. Definition of congruent angles 3. m 2 = m 1 3. Symmetric Property of Equality Definition of congruent angles Flowchart Proof 1 2 m 1 = m 2 m 2 = m Given Definition of congruent angles Symmetric Property of Equality Definition of congruent angles Paragraph Proof 1 is congruent to 2. y the definition of congruent angles, the measure of 1 is equal to the measure of 2. The measure of 2 is equal to the measure of 1 by the Symmetric Property of Equality. Then by the definition of congruent angles, 2 is congruent to hapter 2 Reasoning and Direct Proofs

48 2.6 Exercises Dynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck 1. WRITING Explain why all right angles are congruent. 2. VOULRY What are the two types of angles that are formed by intersecting lines? Monitoring Progress and Modeling with Mathematics In Exercises 3 6, identify the pair(s) of congruent angles in the figures. Explain how you know they are congruent. (See Examples 1, 2, and 3.) F G N M G 50 L H S P 50 J K M Q R K M L W H X Y Z 6. is supplementary to D. D is supplementary to DEF. J In Exercises 11 14, find the values of x and y. (See Example 4.) 11. (8x + 7) 5y (9x 4) 13. (10x 4) (18y 18) (7y 34) 16y 6(x + 2) 12. ERROR NLYSIS In Exercises 15 and 16, describe and correct the error in using the diagram to find the value of x. (13x + 45) (6x + 2) (12x 40) (19x + 3) 4x (7y 12) (6y + 8) (6x 26) 14. 2(5x 5) (5y + 5) (7y 9) (6x + 50) E F D 15. (13x + 45) + (19x + 3) = x + 48 = x = 132 In Exercises 7 10, use the diagram and the given angle measure to find the other three measures. (See Example 3.) 7. m 1 = m 3 = m 2 = x = (13x + 45) + (12x 40) = 90 25x + 5 = 90 25x = 85 x = m 4 = 29 Section 2.6 Proving Geometric Relationships 103

49 17. PROOF opy and complete the flowchart proof. Then write a two-column proof. (See Example 1.) Given Prove l 2, Given Vertical ngles ongruence Theorem (Theorem 2.6) 18. PROOF opy and complete the two-column proof. Then write a flowchart proof. (See Example 2.) Given D is a right angle. E is a right angle. Prove DE D E STTEMENTS 1. D is a right angle. E is a right angle. RESONS and D are complementary. 2. Definition of complementary angles 3. DE and D are complementary DE PROVING THEOREM opy and complete the paragraph proof for the ongruent omplements Theorem (Theorem 2.5). Then write a two-column proof. (See Example 3.) Given 1 and 2 are complementary. 1 and 3 are complementary. Prove and 2 are complementary, and 1 and 3 are complementary. y the definition of angles, m 1 + m 2 = 90 and = 90. y the, m 1 + m 2 = m 1 + m 3. y the Subtraction Property of Equality,. So, 2 3 by the definition of. 104 hapter 2 Reasoning and Direct Proofs

50 20. PROVING THEOREM opy and complete the two-column proof for the ongruent Supplement Theorem (Theorem 2.4). Then write a paragraph proof. (See Example 5.) Given 1 and 2 are supplementary. 3 and 4 are supplementary Prove 2 3 STTEMENTS 1. 1 and 2 are supplementary. 3 and 4 are supplementary m 1 + m 2 = 180, m 3 + m 4 = 180 RESONS 1. Given = m 3 + m 4 3. Transitive Property of Equality 4. m 1 = m 4 4. Definition of congruent angles 5. m 1 + m 2 = 5. Substitution Property of Equality 6. m 2 = m PROOF In Exercises 21 24, write a proof using any format. 21. Given QRS and PSR are supplementary. Prove QRL PSR 23. Given E DE Prove E DE Q P L R S K M N E 24. Given JK JM, KL ML, J M, K L Prove JM ML and JK KL D J K 22. Given 1 and 3 are complementary. 2 and 4 are complementary. Prove M 25. MKING N RGUMENT You overhear your friend discussing the diagram shown with a classmate. Your classmate claims 1 4 because they are vertical angles. Your friend claims they are not congruent because he can tell by looking at the diagram. Who is correct? Support your answer with definitions or theorems L Section 2.6 Proving Geometric Relationships 105

51 26. THOUGHT PROVOKING Draw three lines all intersecting at the same point. Explain how you can give two of the angle measures so that you can find the remaining four angle measures. 28. WRITING How can you save time writing proofs? 29. MTHEMTIL ONNETIONS Find the measure of each angle in the diagram. 27. RITIL THINKING Is the converse of the Linear Pair Postulate (Postulate 2.8) true? If so, write a biconditional statement. Explain your reasoning. 10y (3y + 11) (4x 22) (7x + 4) 30. HOW DO YOU SEE IT? Use the student s two-column proof. Given and 2 are supplementary. Prove STTEMENTS l and 2 are supplementary. RESONS 1. Given 2. m l = m 2 2. Definition of congruent angles 3. m l + m 2 = Definition of supplementary angles 4. m l + m 1 = Substitution Property of Equality 5. 2m 1 = Simplify. 6. m 1 = Division Property of Equality 7. m 2 = Transitive Property of Equality a. What is the student trying to prove? b. Your friend claims that the last line of the proof should be 1 2, because the measures of the angles are both 90. Is your friend correct? Explain. Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Use the cube. 31. Name three collinear points. I 32. Name the intersection of plane F and plane EHG. 33. Name two planes containing. D 34. Name three planes containing point D. 35. Name three points that are not collinear. F G 36. Name two planes containing point J. E J H 106 hapter 2 Reasoning and Direct Proofs

52 2 hapter Review Dynamic Solutions available at igideasmath.com 2.1 onditional Statements (pp ) Write the if-then form, the converse, the inverse, the contrapositive, and the biconditional of the conditional statement leap year is a year with 366 days. If-then form: If it is a leap year, then it is a year with 366 days. onverse: If it is a year with 366 days, then it is a leap year. Inverse: If it is not a leap year, then it is not a year with 366 days. ontrapositive: If it is not a year with 366 days, then it is not a leap year. iconditional: It is a leap year if and only if it is a year with 366 days. Write the if-then form, the converse, the inverse, the contrapositive, and the biconditional of the conditional statement. 1. Two lines intersect in a point. 2. Supplementary angles sum to Right angles are Let p represent You are in the state of Virginia. Let q represent You are in the United States. Write in symbolic form: If you are not in the state of Virginia, then you are not in the United States. Then decide whether the statement is true or false. 2.2 Inductive and Deductive Reasoning (pp ) What conclusion can you make about the sum of any two even integers? Step 1 Look for a pattern in several examples. Use inductive reasoning to make a conjecture = = = = ( 10) = ( 16) = 28 onjecture Even integer + Even integer = Even integer Step 2 Let n and m each be any integer. Use deductive reasoning to show that the conjecture is true. 2n and 2m are even integers because any integer multiplied by 2 is even. 2n + 2m represents the sum of two even integers. 2n + 2m = 2(n + m) by the Distributive Property. 2(n + m) is the product of 2 and an integer (n + m). So, 2(n + m) is an even integer. The sum of any two even integers is an even integer. 5. What conclusion can you make about the product of an even and an odd integer? 6. Use the Law of ontrapositive to make a valid conclusion. If an angle is a right angle, then the angle measures 90. does not measure Use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements: If x = 3, then 2x = 6. If 4x = 12, then x = Write the Law of Detachment in symbolic form. hapter 2 hapter Review 107